Sensitivity analysis of dispersion curves of Rayleigh waves with fundamental and higher modes

Surface wave inversion based on dispersion curves of higher modes is increasingly being applied to estimate near-surface S-wave velocity structures using active or passive sources. Moreover, sensitivity analysis of dispersion curves is important for estimating the confidence of inverted results. In this study, we define a sensitivity function and a sensitivity kernel, give their analytic expressions and then propose a method to estimate the confidence of inverted results and redesign the initial structure through sensitivity analysis. Three kinds of common layered models [velocity-increasing-with-depth (regular), high-velocity layer (HVL), and low-velocity layer (LVL)] are considered. Results show that dispersion curves are more sensitive to S-wave velocity than P-wave velocity and density. The frequency band of the sensitivity function increases as the modes of dispersion curves increase, and the higher modes can penetrate additional depth. For the HVL model, the dispersion curves are particularly sensitive to those of layers above the HVL. By contrast, for the LVL model, the dispersion curves are particularly sensitive to the LVL. Aside from the analysis of sensitivity, the sensitivity functions can also be used for inversion. Thus, we perform an S-wave velocity inversion for synthetic records calculated from the structural model of Shanghai, China. Synthetic noise records are converted into the frequency-velocity domain (c-f panel) by the vector wavenumber transform method. Then, the dispersion curves of multimode are determined from the peaks of the c-f panel. The inverted structure perfectly matches the target structure except the deeper structures that is consistent with the sensitivity analysis of the determined dispersion curves. Finally, we apply the same procedure to the real case in Linyi, Shandong, China. The first three modes of dispersion curves accurately match the data. The synthetic and real examples indicate that the combined use of higher modes offers additional constraint for the velocity structure, which leads to better results of inversion than that of only the fundamental mode. In addition, the sensitivity analysis offers us a tool to estimate the confidence of the inverted structure and redesign the initial structure.